Data Selection for Regression Tasks

For a finite multiset $D=\{z_1,\ldots,z_N\}\subseteq\mathbb R^d$, let

$$L_D(h)=\frac1N\sum_{i=1}^N\|h-z_i\|^2,quad L_D^\star=\min_hL_D(h),$$

and let $L_D^\star(n)$ be the minimum full-data loss obtained by averaging $n$ selected points. More generally, for examples $z=(x,y)\in\mathbb R^d\times\mathbb R^m$, let the min-Frobenius-norm ERM choose a linear map $W\colon\mathbb R^d\to\mathbb R^m$ under loss $\|Wx-y\|^2$.

1. For fixed $d>1$ and $n>1$, what is the value of $F(d,n)=\sup_D L_D^\star(n)/L_D^\star$ for mean estimation in $\mathbb R^d$?

2. For weighted scalar linear regression, what is the value of $\sup_D L_D^\star(n;A^\star)/L_D^\star$ when $d<n<2d$, where $A^\star$ is the min-norm ERM and the learner chooses a convex combination of $n$ example losses?

3. Given $d,m,n$, what is the value of the unweighted general linear-prediction ratio $F(d,m,n)$?

4. Given $d,m,n$, what is the value of the weighted ratio $F_{\rm weighted}(d,m,n)$? In particular, what is the minimal $n=n(d,m)$ such that this ratio equals $1$?